# Boolean ring

In

mathematics, a

**Boolean ring** *R* is a

ring for which

*x*^{2} =

*x* for all

*x* in

*R*; that is,

*R* consists of

idempotent elements. These rings arise from (and give rise to) Boolean algebras. One example is the

power set of any set

*X*, where the addition in the ring is

symmetric difference, and the multiplication is

intersection.

Every Boolean ring *R* satisfies *x* + *x* = 0 for all *x* in *R*, because we know

*x* + *x* = (*x* + *x*)^{2} = *x*^{2} + 2*x*^{2} + *x*^{2} = *x* + 2*x* + *x*

and we can subtract

*x* +

*x* from both sides of this equation. A similar proof shows that every Boolean ring is

commutative:

*x* + *y* = (*x* + *y*)^{2} = *x*^{2} + *xy* + *yx* + *y*^{2} = *x* + *xy* + *yx* + *y*

and this yields

*xy* +

*yx* = 0, which means

*xy* = −

*yx* =

*yx* (using the first property above).

If we define

*x* &and *y* = *xy*,

*x* ∨ *y* = *x* + *y* − *xy*,

- ~
*x* = 1 + *x*

then these satisfy all of the axioms for meets, joins, and complements in a Boolean algebra. Thus every Boolean ring with 1 becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring with 1 thus:

*xy* = *x* ∧ *y*,

*x* + *y* = (*x* ∨ *y*) ∧ ~(*x* ∧ *y*).